Asensio, 26, is out of contract with Madrid at the end of the season. While he has not ruled out extending his eight-year stay at the club, there are no renewal talks.
"Honestly, I don't know if they have called," Asensio told Spanish radio station Cope when asked if he was aware of Barca's interest.
"A lot of other clubs have been linked [with me] as well. But it's normal because in seven months, well, from January actually, I am free to speak to other teams."
Asensio added that the links to Barca had led to him receiving "over 200 messages" from friends wanting updates on his future.
Pressed on if he could rule out switching Clasico clubs, like other Madrid and Barca players have done in the past, the former Mallorca player would not give a definitive response.
"It's just that I have not thought about [joining Barca] or considered it," he said. "Honestly, right now, I can't give you an answer. I don't know."
Barca also made an attempt to sign Asensio when he was coming through at Mallorca, but he eventually opted to join Madrid.
He has made 239 appearances for Los Blancos, scoring 50 goals, and won three league titles and three Champions Leagues, among other trophies.
However, Asensio has fallen down the pecking order at the club recently, with Federico Valverde and Rodrygo among those picked ahead of him, although he said his relationship with coach Carlo Ancelotti remains good.
"August was tough with the [uncertainty over my future], but the decision to stay [this season] has been made and I am happy to be at Madrid," he added.
"Ancelotti's confidence in me was key. I still believe I can help Madrid and I want to keep winning trophies. There were options [to leave in the summer] but I never really thought I would leave and Ancelotti was part of the reason why."